Czech Technical University in Prague

Faculty of Electrical Engineering

MASTER THESIS

Denis Efremov

Unstable ground vehicles and artificial stabilitysystems

Department of Control Engineering

Supervisor of the master thesis: Doc. Ing. Martin Hromcık, Ph.D.

Study programme: Cybernetics and Robotics

Specialization: Systems and Control

Prague 2018

ZADÁNÍ DIPLOMOVÉ PRÁCE

I. OSOBNÍ A STUDIJNÍ ÚDAJE

420311Osobní číslo:DenisJméno:EfremovPříjmení:

Fakulta elektrotechnickáFakulta/ústav:

Zadávající katedra/ústav: Katedra řídicí techniky

Kybernetika a robotikaStudijní program:

Systémy a řízeníStudijní obor:

II. ÚDAJE K DIPLOMOVÉ PRÁCI

Název diplomové práce:

Nestabilní pozemní vozidla a systémy pro jejich aktivní stabilizaci

Název diplomové práce anglicky:

Unstable ground vehicles and artificial stability systems

Pokyny pro vypracování:The goal of the thesis is to investigate the flight-controls results of the 1980's related to statically unstable aircraft equippedwith artificial stabilization systems (CCV, control-configured vehicles), and investigate their applicability for design ofhighly-manevourable high-performance cars.1. Adopt suitable high-fidelity and low-fidelity lateral-dynamics simulation models.2. Prepare variants with high, reduced, neutral, and lost stability.3. Realize simulated ride tests on the simulation platform.4. Investigate and design suitable control laws for lateral dynamics artificial stabilization, and realize related simulated ridetests.5. Investigate estimation algorithms that can be useful to compensate for missing essential measurements, namely thetyre-models-based observers for estimation of the sideslip angle.

Seznam doporučené literatury:[1] Edward M. Kasprzak, L. Daniel Metz, William F. Milliken Douglas L. Milliken, Race Car Vehicle Dynamics - Problems,Answers and Experiments, Premiere Series Books, 2015, ISBN-10: 0768011272.[2] Sigur Skogestad, Ian Postletwaite, Multivariable feedback control, Wiley, 2015, ISBN-10: 0470011688.

Jméno a pracoviště vedoucí(ho) diplomové práce:

doc. Ing. Martin Hromčík, Ph.D., katedra řídicí techniky FEL

Jméno a pracoviště druhé(ho) vedoucí(ho) nebo konzultanta(ky) diplomové práce:

Termín odevzdání diplomové práce: 25.05.2018Datum zadání diplomové práce: 15.02.2018

Platnost zadání diplomové práce: 30.09.2019

_________________________________________________________________________________prof. Ing. Pavel Ripka, CSc.

podpis děkana(ky)prof. Ing. Michael Šebek, DrSc.

podpis vedoucí(ho) ústavu/katedrydoc. Ing. Martin Hromčík, Ph.D.

podpis vedoucí(ho) práce

© ČVUT v Praze, Design: ČVUT v Praze, VICCVUT-CZ-ZDP-2015.1

Acknowledgements

I would like to thank my supervisor doc. Ing. Martin Hromcık, Ph.D. for his valu-

able advice, ideas, and support during the creation of this master thesis.

Great thanks also belong to my colleges: Bc. Vıt Cibulka for excellent help with

the simulator; Bc. Marek Laszlo for his critics, which helped during this work. Also,

I would like to thank my opponent, Ing. Martin Mondek, without his advice and his

master thesis this work would never exist.

Mainly I would like to thank my family for invaluable help with my education, and

my girlfriend and friends, without whom this thesis would not be completed.

I hereby declare that I have completed this thesis with the topic ”Unstable ground

vehicles and artificial stability systems” independently and that I have included a full

list of used references. I have no objection to the usage of this work in compliance with

the act §60 Zakon c.121/2000 Sb. (copyright law).

In ................. date ................. signature of the author

Abstrakt:

Cılem teto diplomove prace je definovat pozemnı vozidlo s nestabilnı bocnı dy-

namikou pomocı nazvoslovı zavedeneho Spolecnostı automobilovych inzenyru, porov-

nat klady a zapory nestabilnıch aut, prodiskutovat moznosti jejich stabilizace pomocı

systemu aktivnıho rızenı zalozeneho na sledovanı reference na rychlost zmeny smeru.

Na konci teto prace jsou porovnavany ruzne rıdicı prıstupy, jsou diskutovany algoritmy

pro odhad skluzu a silova zpetna vazba na volant.

Klıcova slova: Nestabilnı Pozemnı Vozidlo; Umele Stabilizacnı Systemy; Sledovanı

Rychlosti Zmeny Smeru; Pretaciva Auta; Silova Zpetna Vazba; Odhad Bocnıho Skluzu;

Odhad Skluzu Pneumatiky; Jednostopy Model; Pacejka Magic Formula; Steer-by-Wire.

Abstract:

This master thesis aims to define laterally unstable ground vehicle regarding the

nomenclature introduced by Society of Automotive Engineers, lists pros & cons of

making cars to be unstable, to discuss the possibilities of their stabilization by active

steering system based on yaw rate tracking control architectures. At the end of this

work, different control approaches are compared, slip estimation algorithms and force

feedback are discussed.

Keywords: Laterally Unstable Ground Vehicle; Artificial Stabilization Systems; Yaw

Rate Tracking; Oversteering Cars; Force Feedback; Side-Slip Estimation; Tire’s Slip

Estimation; Singel-Track Model; Pacejka Magic Formula; Steer-by-Wire.

Contents

1 Introduction 21.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Objectives 4

3 Used Vehicle Modelling Approaches 53.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Nonlinear Single-Track Model . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2.1 Used assumptions and simplifications . . . . . . . . . . . . . . . 53.2.2 Steering angle projection and vehicle dynamics . . . . . . . . . . 63.2.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.4 Block representation . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Linearized Single-Track Model . . . . . . . . . . . . . . . . . . . . . . . 83.4 Tyre Modelling Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4.1 Pacejka Magic Formula . . . . . . . . . . . . . . . . . . . . . . . 103.4.2 Two-Lines Tire Model . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Laterally Unstable Vehicle 134.1 Definition of Laterally Unstable Vehicle . . . . . . . . . . . . . . . . . . 134.2 Critical Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Practical Meaning of Lateral Instability . . . . . . . . . . . . . . . . . . 174.4 Pros & Cons of Unstable Vehicles . . . . . . . . . . . . . . . . . . . . . . 17

5 Proposed Control Architectures 205.1 Yaw Rate Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Augmented Yaw Rate Tracking: Side-slip and Steering Angle Control . 215.3 Augmented Yaw Rate Tracking: Tire’s Slip Sum . . . . . . . . . . . . . 225.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Simulation Tests 256.1 Yaw Rate Tracking Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Spoiler Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Slip Estimation Algoritms 307.1 Overview of Tire’s Slip Angle Estimators . . . . . . . . . . . . . . . . . 307.2 Overview of Side-Slip Angle Estimators . . . . . . . . . . . . . . . . . . 31

8 Force Feedback 338.1 Overview of Steer-by-Wire Vehicles . . . . . . . . . . . . . . . . . . . . . 338.2 The Implementation Used in This Thesis . . . . . . . . . . . . . . . . . 33

9 Results 36

10 Conclusion 37

11 Bibliography 38

List of Figures 40

1

1. Introduction

According to the Oxford dictionary, science is ”the intellectual and practical activi-

ty encompassing the systematic study of the structure and behavior of the physical and

natural world through observation and experiment.” A scientist is always a romantic

man, who answers the most interesting question in the world: ”What if?”

After the design of the first successful airplane made by the Wright brothers, many

scientists and engineers were charged to invent faster and agiler aircraft driven by the

question: ”What if?”

During the first half of the 20th century, the physics behind the airplanes have

been described. The equations of flight have been derived. Researchers arrived to an

understanding of the weight of each parameter in these equations. They were trying to

answer the question: ”What if?”

These attempts have borne their fruits. In 1980’s the first statically unstable air-

craft equipped with artificial stabilization systems took off in the sky. It was the

control-configured version of Jaguar fighter, in witch regulators were used to stabilize

the aircraft at supersonic speeds, where it is unstable [16]. The epoch of the new

generation of agiler fighters began. Because of the most interesting question for every

scientist: ”What if?”

This thesis is inspired by results of that works behind the unstable aircraft and

trying to answer the question: ”What if?” Namely, ”What if we create a laterally

unstable ground vehicle?”

1.1 Outline

This work is divided into ten parts.

In the first two parts, which are [Introduction] and [Objectives], work descrip-

tion and goals are stated.

The part [Used Vehicle Modelling Approaches] introduces used vehicle’s and

tire’s modelling techniques.

In part, which is named [Laterally Unstable Vehicle], the laterally unstable ve-

hicle defined and its pros and cons are listed.

The 5th chapter, which is [Proposed Control Architectures], describes the pos-

sible control architectures which can be used to stabilize the laterally unstable ground

2

vehicle.

Chapter [Simulation Tests] shows the simulation results and comparison of the

proposed control architectures.

The next two chapters are presented here just for completeness of the work because

it was not a primary purpose of the thesis. The sixth part, which is named [Slip Esti-

mation Algoritms], provides a short overview of the slip angle estimation algorithms.

In part [Force Feedback], the steer-by-wire vehicle is overviewed. Also, a force

feedback design implemented in the simulator is shown.

The next part, [Results], lists reached goals.

The last part, [Conclusion] summarises this thesis and proposes possible future

works in the field of laterally unstable ground vehicles.

3

2. Objectives

The primary objectives of this thesis are:

• Adopt suitable lateral-dynamics simulation model

• Define lateral instability of a ground vehicle

• Create a simulator based platform

• Investigate possible control techniques for a laterally unstable ground vehicle

• Realize simulated ride tests

• Investigate estimation algorithms that can be useful to compensate for missing

essential measurements

4

3. Used Vehicle Modelling

Approaches

3.1 Introduction

This chapter describes used vehicle modeling approaches and tire models. The pro-

posed models were used to analyze a lateral vehicle dynamics and to compare different

control techniques.

The nonlinear single-track model is derived in the section [Nonlinear Single-

Track Model], its liniarization is provided in the next section [Linearized Single-

Track Model], the last section [Tyre Modelling Approaches] describes the used

tire modeling techniques.

3.2 Nonlinear Single-Track Model

For vehicle models there exist two main modeling techniques, which are mainly used

to describe the vehicle dynamics: single-track [1, chapter 9] and twin-track models [1,

chapter 10]. During this work, the first one was used, because it has sufficient fidelity

for describing a lateral vehicle dynamics, used in a huge amount of articles touching

lateral vehicle dynamics [2,3,4,5] and it is simpler than the second one (three versus

sixteen states).

The same modeling technique was used in work of my predecessor, M. Mondek in

[6]. But his model derivation begins with a creation of an equilibrium of all forces

acting on a vehicle’s center of gravity along x and y axis and moments acting around

the z axis. In this work, the derivation begins with the derivation of equations of lat-

eral, longitudinal and yaw motion. Also, here all nonlinearities are abandoned in the

state space representation (Eq. 3.6). Despite differences in nonlinear models, resulting

linearized models are the same.

3.2.1 Used assumptions and simplifications

The nonlinear single-track model is used to describe planar vehicle motion. The

derivation of the model is used in the following articles: [2,3]. Used simplifications

during the derivation have been [1]:

• All lifting, rolling, and pitching motion is neglected.

• Vehicle mass is assumed to be concentrated at the center of gravity.

5

• Front and rear tires are represented as one single tire on each axle. Imaginary

contact points of tires and surface are assumed to lie along the center of axles.

• Pneumatic trail and aligning torque resulting from a side-slip angle of a tire are

neglected.

• Mass distribution on the axles is assumed to be constant.

• Longitudinal forces on tires, resulting from a normalized tire slip angle, are ne-

glected. All longitudinal forces acting on each axle are assumed to be strictly

from the engine.

3.2.2 Steering angle projection and vehicle dynamics

Vehicle coordinate system used in this thesis is shown in Figure [3.1]. It is the con-

ventional right-hand Cartesian coordinate system. The x axis follows from the center

of gravity to the front of the vehicle. The y axis goes towards the left side of the car,

and z axis lies from the center of gravity to the top of the vehicle. Vehicle’s yaw has

positive angle increment while turning to the left.

ψZ

XY

Figure 3.1: The vehicle coordinate system

The single-track model followed the assumptions above is presented in Figure [3.2].

Here, three degrees of freedom exist:

• longitudinal motion

−mv(β + ψ) sinβ +mv cosβ = Fx; (3.1)

6

Fy,r

Fx,r

Fx,f

Fy,f

vvf

vr

β αf δ

αr ψ

lr lfFigure 3.2: The single-track model

• lateral motion

−mv(β + ψ) cosβ +mv sinβ = Fy; (3.2)

• yaw motion

Izψ = Mz. (3.3)

Here, m is the vehicle’s mass, v is the velocity of the center of gravity of the vehicle,

β is the side-slip angle, ψ is the yaw angle, Iz is the moment of inertia of the vehicle

around the z axis. On the right-hand side of the equations, there are forces acting on

the center of gravity of the vehicle along with x (Fx) and y (Fy) axes and the moment

acting around the z axis (Mz).

From the equations above the following matrix equation can be written down:mv(β + ψ)

mv

Izψ

=

− sinβ cosβ 0

cosβ sinβ 0

0 0 1

Fx

Fy

Mz

. (3.4)

The forcers Fx and Fy together with the moment Mz can be described by the

following steering angle projection (also called the force coordinate transformation):

Fx

Fy

Mz

=

cos δ − sin δ 1 0

sin δ cos δ 0 1

lf sin δ lf cos δ 0 −lr

Fxf

Fyf

Fxr

Fyr

, (3.5)

where δ is the steering angle, lf is the distance from the vehicle’s center of gravity

to the front axle (the front wheel), and lr is the distance from the vehicle’s center of

gravity to the rear axle (the rear wheel). Forces Fyf and Fyr are calculated via Pacejka

7

Magic formula described in the section [Tyre Modelling Approaches]. During this

work, the vehicle is assumed to be Rear-wheel drive (RWD), so Fxf = 0 and Fxr is the

second input into the system after the steering angle δ.

After simplifications above the resulting system of nonlinear differential equations

describes the steering angle projection and vehice dynamics can be written as follows:

β = −ψ + 1mv (cosβ(Fyr + cos δFyf )− sinβ(Fxr − sin δFyf )),

v = 1m(sinβ(Fyr + cos δFyf ) + cosβ(Fxr − sin δFyf )),

ψ = 1Iz (lf cos δFyf − lrFyr).

(3.6)

The system (Eq. 3.6) has two inputs: steering angle δ and force acting on rear tire

along x axis Fxr. Also, it has three states: the side-slip β, vehicle’s centre of gravity

velocity v, and it’s yaw rate ψ.

3.2.3 Kinematics

The kinematics of the single-track vehicle model includes calculation of the tires’

side-slip angles. Those angles are used in Pacejka Magic formula for calculation of the

side tires’ forces Fyr and Fyf . The calculation of tires’ sleep angles takes into accaunt

the steering angle δ, side-slip angle β, velocity of the center of gravity of the vehicle

v, and the yaw rate ψ as (formulas are taken from dimploma thetis made by Martin

Mondek [6]:

αf = δ − arctan

(v sinβ + lf ψ

v cosβ

), (3.7)

αr = − arctan

(v sinβ − lrψv cosβ

), (3.8)

where αf and αr are the tire’s side-slip angles of the front and rear tires respectively.

3.2.4 Block representation

The block diagram of the nonlinear singe-track model is shown in Figure [3.3].

3.3 Linearized Single-Track Model

Linearization of the single-track model (Eq 3.6) is provided assuming that steering

angle δ and side slip β are supposed to be smaller than 10 degrees, thus the first

assumption can be used:

sinx ≈ x, cosx ≈ 1. (3.9)

8

FxrSteering angleprojection  Dynamics

δ

Tire models Kinematics

v

β

ψ˙

αf

αr

Fyf

Fyr

Fx

Fy

Mz

Figure 3.3: The block diagram of the single-track model

With Eq 3.9, the equation 3.6 takes the following form:mv(β + ψ)

mv

Izψ

=

−β 1 0

1 β 0

0 0 1

Fx

Fy

Mz

. (3.10)

The velocity is assumed to be constant, thus its derivation is zero v = 0. The second

row yeilds Fx = −βFy and with β2 � 1 rad. [11] we got:(mv(β + ψ)

Izψ

)=

(Fy

Mz

). (3.11)

The tire slip angles αf and αr can also be linearized as it’s done in [6]:

αf = δ − β −lf ψ

v, (3.12)

αr = −β +lrψ

v. (3.13)

Then lateral forces acting on each tire Fyf and Fyr can be estimated by linear tire

model (Eq. 3.24) as:

Fyf = Cfαf = Cf

(δ − β −

lf ψ

v

), (3.14)

Fyr = Crαr = Cr

(−β +

lrψ

v

), (3.15)

where Cf and Cr are cornering stifnesses of the front and rear tires respectivelly.

Now, the lateral forces acting on the vehilce’s centre of gravity can be derived from

9

Eq. 3.5 assuming Fxf = 0 (the car is assumed to be RWD above) and Eq. 3.9 as:

Fy = Fyr + Fyf . (3.16)

Finally, using all assumptions above, the Eq 3.6 takes the following form:

β = −Cf + Crmv

β +

(lrCr − lfCf

mv2− 1

)ψ +

Cfmv

δ, (3.17)

ψ =lrCr − lfCf

Izβ −

l2fCf + l2rCr

vIzψ +

lfCfIz

δ. (3.18)

The state-space representation now can be written in matrix form as:

(β

ψ

)=

−Cf+Cr

mvlrCr−lfCf

mv2− 1

lrCr−lfCf

Iz− l2fCf+l

2rCr

vIz

(βψ

)+

Cf

mv

lfCf

Iz

δ. (3.19)

Now, the transfer function from the steering wheel δ to the vehicle’s yaw rate ψ is

derived as:

Gψ(s) =ψ

δ=

b1s+ b0s2 + a1s+ a0

, (3.20)

where coefficients a0, a1, b0, b1 are:

b0 =CfCr(lf + lr)

Izmv, b1 =

Cf lfIz

, (3.21)

a0 =CfCr(lf + lr)

2

Izmv2+Crlr − Cf lf

Iz, a1 =

Cr + Cfmv

+Crl

2r + Cf l

2f

Izv. (3.22)

The derived transfer function is the same as in [4], this fact allows using conclusion

made in the article above, what is provided in the Chapter [Laterally Unstable

Vehicle].

3.4 Tyre Modelling Approaches

The tire dynamics have a significant impact on the whole vehicle dynamics [7].

There exist a lot of different techniques, which are used to estimate forces and torques,

which are generated by tires. Those models are correctly described and compared by

Lukas Haffner in [7]. In this work, only two models described below are used.

3.4.1 Pacejka Magic Formula

The most famous tire model is proposed by Hans Bastiaan Pacejka [8], and it consid-

ers more than 20 different coefficients. For simplification, he introduced the Simplified

Pacejka Magic formula [7], which can be used for estimation not only the the lateral

and longitudinal forces’ impact on a tire, but also all the torques acting on a wheel

around all axis. It has a straightforward calculation, and the same formula is used to

10

estimate all the forces and torques using different sets of coefficients.

The general Simplified Pacejka Magic formula has the following equation:

F (α) = DFz sin(C arctan(Bα− E(Bα− arctan(Bα)))), (3.23)

where D, C, B, and E is the set of shaping coefficients, Fz is a wheel-load, α is a tire’s

slip-angle (in case of Fx the normalized slip is used), F is an estimated force or torque.

As it is mentioned above, in section [Nonlinear Single-Track Model], in this

work only the lateral forces Fy (Figure [3.4]) produced by tires are considered. And, as

it is described in the chapter [Force Feedback], the aligning torque Mz (Figure [3.5])

is used for calculating the force feedback applied by simulator on the steering wheel.

-15 -10 -5 0 5 10 15-5000

0

5000F

z = 1000 [N]

Fz = 2000 [N]

Fz = 3000 [N]

Fz = 4000 [N]

Fz = 5000 [N]

Figure 3.4: Dependence of generated lateral force Fy on tire slip-angle α with differentloads Fz

3.4.2 Two-Lines Tire Model

In [12] a simple tire model is utilized. This model contains three parts: two hori-

zontal saturations and one linear part which is a linear approximation of the tire force

around zero tire slip angle (or slip ratio in case of longitudinal force)(Figure 3.6). The

model equation looks like:

F =

Cα |α| < µmax

C ,

µmax |α| ≥ µmax

C .(3.24)

The parameter µmax is a maximal friction coefficient, C is a cornering stiffness in

case of Fy (or tire slip coefficient in case of Fx). The equation works for a tire’s slip

angle α in case of a lateral force Fy or for a tire’s slip ratio λ in case of longitudinal

force Fx.

11

-15 -10 -5 0 5 10 15-500

0

500

Fz = 1000 [N]

Fz = 2000 [N]

Fz = 3000 [N]

Fz = 4000 [N]

Fz = 5000 [N]

Figure 3.5: Dependence of generated aligning moment Mz on tire slip-angle α withdifferent loads Fz

-15 -10 -5 0 5 10 15-5000

0

5000

PacejkaTwo-Lines

Figure 3.6: Compirison of Pacejka and Two-Lines tire models

12

4. Laterally Unstable Vehicle

This chapter answers a question: ”What is exactly an unstable car?” And the fol-

lowing question: ”What it exactly brings to automotive and what is a price?”

The chapter is separated into 4 sections: the section [Definition of Laterally

Unstable Vehicle] defines the unstable car via existing nomenclature used in auto-

motive; the section [Critical Velocity] describes the key parameter of an unstable car;

the section [Practical Meaning of Lateral Instability] shows the physics behind

this term, and the last section [Pros & Cons of Unstable Vehicles] summarises the

positive and negative factors of making vehicles to be laterally unstable.

4.1 Definition of Laterally Unstable Vehicle

According to Society of Automotive Engineers (SAE) [13], in vehicle dynamics,

there are two terms that describe the sensitivity of a car to steering: understeering and

oversteering.

Undesteer occurs when a car steers less than angle commanded by the driver, on

the other hand, oversteer occures in the case when a car steers more than amount

commanded by the driver. Both cases are defined for steady-state cornering where the

vehicle follows the constant-path at a constant speed and with a constant steering angle,

on a flat surface. The comparison of understeer and oversteer car is shown in Figure 4.1.

The precise definition in [13] is provided via understeer gradient K as:

• when K > 0 a vehicle is understeering;

• when K = 0 a vehicle is neutral steering;

• when K < 0 a vehicle is oversteering.

According to [14] the understeer gradient is defined as a derivative of the front tires’

average steer angle with respect to the lateral acceleration imposed to the vehicle at

its center of gravity. Figure 4.2 shows an example of its dependency. Reference to [14],

the understeer gradient can be calculated as:

K =

(lrCf−lfCr

)m

2(lf + lr), (4.1)

where lr (lf ) is distance from rear (front) axle to the vehicle’s center of gravity; Cr

(Cf ) is a rear (front) tire’s cornering stifness coefficient, and m is vehicle’s mass.

13

(a) Understeering car cornering. (b) Oversteering car cornering.

Figure 4.1: Comparison of understeering and oversteering car. Images are taken fromhttps://en.wikipedia.org. Green path is an optimal cornering trajectory. Understeer:the car goes not turn enough and leave the road. Oversteer: the car turns more sharplyand could get to spin.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

8

9UndersteerNeutralOversteer

Figure 4.2: Understeer and oversteer definition

It is obvious that m2(lf+lr)

is always positive. Thus the sign of the understeer gradient

K depends only on the sign of the equation:

sign(K) = sign(lrCr − lfCf ). (4.2)

Therefore, in this work the following definitions are used:

• A vehicle is understeering when lrCr − lfCf > 0;

• A vehicle is neutral steering when lrCr = lfCf ;

• A vehicle is oversteering when lrCr − lfCf < 0.

According to [4], Long Wang and Jurgen Ackermann call understeering cars in

manner of automatic control engineers as always stable (means that poles of the open

loop of the Eq. 3.20 are on the left-half plane for all possible velocities of the vehicle’s

14

center of gravity v > 0); and oversteering cars as unstable, because those transfer

functions can have unstable poles, and it depends on vehicle’s velocity (its dependences

is discussed in the next section [Critical Velocity]). In this work understeering vehicle

can be named as stable vehicle and oversteering as unstable. The neutral steering car

can be named as metastable.

4.2 Critical Velocity

According to [4] and [6], the understeering car is always stable, and it has no re-

striction on critical velocity, because of parameter a0 from Eq. 3.20 is always greater

then 0 in case when lrCr − lfCf > 0. The Figure 4.3 shows changing poles’ location of

Eq. 3.20 regarding to different velocities in case of understeering car.

5 ms-1

100 ms-1

Figure 4.3: Poles location of Eq. 3.20 for understeering car depended on differentvehicle’s velocities. Arrows show the direction of poles allocation with higher velocity.Car’s parameters: lf = 1.11 m, lr = 1.89 m, Cf = 138820 N·rad−1, Cr = 236620N·rad−1, m = 1190 kg, Iz = 1141 kg·m−2.

But, is a case of oversteering car, when lrCr − lfCf < 0, the a0 can be less than 0,

what sets one pole to the right-half plane, and a vehicle becomes unstable on higher

velocities. The border of stability of such car is defined as a critical speed vcrit. An

oversteering car has unstable lateral characteristics when its velocity of the center of

gravity v is higher than its critical velocity, which is defined as:

vcrit = (lf + lr)

√CfCr

m(Cf lf − Crlr). (4.3)

The Figure 4.4 ilustrates the changing poles’ location of the transfer function (Eq.

3.20) for a laterally unstable car. The example of stability and instability of the same

vehicle is presented in Figure 4.5, where its path on different velocities is shown. When

the car goes over its critical velocity, it gets to spin.

15

5 ms-1 100 ms-1

Figure 4.4: Poles location of Eq. 3.20 for oversteering car depended on different vehicle’svelocities. Arrows show the direction of poles allocation with higher velocity. Car’sparameters: lf = 2.07 m, lr = 0.93 m, Cf = 258700 N·rad−1, Cr = 116730 N·rad−1,m = 1190 kg, Iz = 3900 kg·m−2. In this case vcrit ≈ 23 m·s−1.

-20 -15 -10 -5 0 5 10 15 20 25-5

0

5

10

15

20

25

30v = 10 [m/s]v = 30 [m/s]

Figure 4.5: Paths of the same vehicle on the different velocities, steering angle δ = 0.2rad. Cars are plotted every 1.5 s. Car’s parameters are the same as in the Figure 4.4.In this case vcrit ≈ 23 m·s−1. (For the path generation a plotting function created byM.Mondek in [6] was used.)

More about critical speed and its dependencies on all parameters can be seen in [6].

In provided tests, there appeared an issue with the defference of theoretically calcu-

lated critical velocity derived within analyzes of the linear model and critical velocity

observed on the nonlinear model. These velocities do not match and typically, the the-

oretical calculated value is greater than value estimated by observation (vcrit,lin ≈ 23

m·s−1 versus vcrit,nonlin ≈ 12 m·s−1 in case of the car with parameters from the Figure

[4.4]). This fact leads to using robust design methods for creating regulators for real

ground vehicles. This statement must be checked in future works.

16

4.3 Practical Meaning of Lateral Instability

The main difference between understeering and oversteering cars is that during cor-

nering maneuver the first one has grater slip angle’s on both tires. Hence, the practical

meaning of lateral instability is pretty simple: unstable vehicle doing a cornering maneu-

ver at a velocity greater than its critical speed can get a spin. This is the main problem

and general reason, why automotive companies are producing common consumer cars

as understeering. Let it be less agile, but don’t let it get a spin doing lateral maneuvers.

The Figures 4.6 and 4.7 show a comparison of tires’ slip angels of the same unstable

vehicle at different velocities. It can be seen that when an unstable vehicle goes on

velocity greater than critical, the sum of slip angles of both tires grows dramatically.

In comparison, when the same car goes on velocity less than its critical, the sum is

αf + αr ≈ 0.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

f+

r

f

r getting spin

Figure 4.6: Tires’ slip angles during cornering maneuver v = 25 m s−1, δ = 0.08 rad(≈ 4.5 deg) of unstable vehicle with parameters as in Figure 4.4

4.4 Pros & Cons of Unstable Vehicles

There are several profits of making vehicles laterally unstable, however, there are

costs to be paid. All of the pros and cons of oversteering cars are listed below.

Laterally unstable vehicles have the following pros:

• An oversteering car is agiler: in steady-state cornering maneuver, it always will

have yaw rate higher than the same but understeering car. The Figure 4.8 visu-

17

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.02

0

0.02

0.04

0.06

0.08

f+

r

f

r

Figure 4.7: Tires’ slip angles during cornering maneuver v = 10 m s−1, δ = 0.08 rad(≈ 4.5 deg). Vehicle parameters are the same as in Figure 4.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

GC at 37% of LGC at 42% of LGC at 50% of LGC at 55% of L

Figure 4.8: Yaw rate of the vehicle with m = 1190 kg, L = lf + lr = 3 m. Steeringangle δ = 0.01 rad., velocity v = 30 m·s−1. Cars with GC (gravity center) equals to37, and 42 percent are understeering; 50 - neutral steering; and 55 - oversteering.

alizes yaw rate of the same vehicle (with the changed center of gravity location)

doing the same cornering maneuver.

• Rear wing used for lateral stabilization on higher velocities can be eliminated.

That reduces the aerodynamic drag of a vehicle and as the primary result, it

saves energy to spend on its drag. The chapter [Simulation Tests] overviews

among other things, this option.

• And the main advantage is that as it is shown in [14], yaw rates feedback to

the front wheel steering angle by a positive PID controller cannot destabilize

oversteering cars below their critical velocity. At the same time, for understeering

cars, there is a possibility of destabilization (condition is derived in [14]). This

feedback is the central part of all proposed regulators listed below in the chapter

18

[Proposed Control Architectures].

Lateral instability brings the following cons:

• The necessity of regulation of the whole steering system arises [4].

• For part of the proposed regulators, tire’s slip angles αf and αr or side-slip β

should be estimated, because there is no possibility of direct measure exists. The

overview of existing estimators is presented in the chapter [Slip Estimation

Algoritms].

• Using regulation, we get a steer-by-wire vehicle. This problematic is considered

below in the chapter [Force Feedback].

19

5. Proposed Control

Architectures

This chapter describes proposed regulation architectures, which are then tested in

the chapter [Simulation Tests]. Root locus characteristics of all controllers are shown

on a vehicle with parameters from the Figure 4.4, which is slightly unstable and has

critical velocity vcrit ≈ 50 m·s−1.

Section [Yaw Rate Tracking] describes basis control approach, sections [Aug-

mented Yaw Rate Tracking: Side-slip and Steering Angle Control] and [Aug-

mented Yaw Rate Tracking: Tire’s Slip Sum] show its possible augmentation.

Section [Conclusion] lists possibilities for making the regulation better.

5.1 Yaw Rate Tracking

This control technique is a basis for each type of the following regulators. It is based

on the idea described by Long Wang and Jurgen Ackermann in [11]. In this article, the

possibility of control by positive PID regulator on feedback from the yaw rate to the

steering angle is described. Especially, they conclude that this type of regulator cannot

destabilize the oversteering car moving over its critical velocity.

The transfer function from the steering angle to the yaw rate is (Eq 3.20). The

diagram in Figure 5.1 shows the regulator signal flow. On the integrator suitable anti-

wind-up saturation should be applied. The whole output goes through saturation block

which means maximum possible steering angle is set to the front axle.

1

Reference

1

SteeringAngleSaturation1

s

Integrator

-K-

P

-K-

I

2

YawRate

Figure 5.1: Diagram of yaw rate tracking control.

For creating the regulator, root locus technique was used. The Figure 5.2 presents

the root locus of a closed loop system for an oversteering car with PI controller. In

Figure 4.4 the location of the poles of an open loop system is presented. As we can

see, there is no oscillation characteristic. Therefore, in a closed loop, poles are set to

the real axis to save vehicle’s native dynamics.

20

-50 -45 -40 -35 -30 -25 -20 -15 -10-1

-0.5

0

0.5

1PolesZeros

Figure 5.2: Root locus of the closed system at v = 50 m·s−1.

5.2 Augmented Yaw Rate Tracking: Side-slip and Steer-

ing Angle Control

This control technique uses yaw rate reference and current velocity to calculate

references for side-slip and steering angles, and positive P regulators to reach the ref-

erences in internal loops. Experimental simulation tests showed that the anti-wind-up

saturations for the external yaw rate tracking loop can be compromised.

For the calculation of the set points for side-slip and steering angles linearized

single-track model (Eq. 3.19) was used:

δref =a11a22ψref − a21a12ψref

a21b1 − a11b2, (5.1)

βref =b1δref − a12ψref

a11, (5.2)

where ai,j , i, j ∈ {1, 2} are coefficients of the matrix A, and bi, i ∈ {1, 2} are coeffi-

cients of the matrix B of the linearized single-track model (Eq. 3.19).

The transfer function from the steering angle to the side-slip angle is:

G(s) =β(s)

δ(s)=

b1s+ b0s2 + a1s+ a0

, (5.3)

where coefficients a0, a1, b0, b1 are:

b0 =CfCrlr(lf lr)

mv2Iz− 1, b1 =

Cfmv

, (5.4)

a0 =CfCr(lf + lr)

2

Izmv2+Crlr − Cf lf

Iz, a1 =

Cr + Cfmv

+Crl

2r + Cf l

2f

Izv. (5.5)

21

The Figure 5.3 presents the changing of pole location within introducing of the

closed loop with feedback from the steering angle to side-slip angle.

Figure 5.3: Changing of pole location by side-slip damper for unstable car at v = 50m·s−1.

The Figure 5.4 shows a diagram of signal flow in the controller.

1

Ref for YawRate

2

YawRate

1

SteeringAngle

3

Velocity

-K-P

-K-P

Saturation

YawRateRef

Velocity

DeltaRef

BetaRef

Ref for Side-Slip and Steering angles

1s

Integrator

-K-

Gain2

-K-

Gain3

P

I

Figure 5.4: Diagram of augmented yaw rate tracking control with the addition of side-slip and steering angles control.

5.3 Augmented Yaw Rate Tracking: Tire’s Slip Sum

This control technique uses the sum of tire’s slip angles for creating of an artifi-

cial border for maximum possible yaw rate on high velocities. The simulation model

equipped with this type of controller did not get to a spin situation (that caused to

other kinds of regulation on a higher speed with keeping a reference for yaw rate on a

physical maximum of tested cars). The primary problem of that control architecture

is regulation delay about 0.2-0.3 s.

The Figure 5.5 shows an example of bounding on high velocity. In comparison, in

Figure 5.6, the behavior of the same control technique on smaller speed is presented.

22

Also, the small delay in reference tracking can be seen.

6 8 10 12 14 16 18 20 22 24 26-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 5.5: The behavior of the regulated unstable car equipped with yaw rate trackingwith tire’s slip sum damper on velocity v = 50 m·s−1. For higher yaw rate referencesartificial boundness arises.

5 10 15 20-1

-0.5

0

0.5

1

Figure 5.6: The behavior of the regulated unstable car equipped with yaw rate trackingwith tire’s slip sum damper on velocity v = 20 m·s−1. There is no bounding.

The transfer function from the steering angle to tire’s slip angles sum is derived

from linearized tire’s slip angles (Eq. 3.12 and Eq. 3.13):

Gα+(s) =(αf + αr)(s)

δ(s)= 1(s)− 2Gβ(s) +

lr − lfv

Gψ(s), (5.6)

The Figure 5.7 presents root locus of the internal closed loop. As can be admitted,

it has an unstable zero and pole. Nevertheless, the outer loop keeps the system in the

stable zone.

23

Figure 5.7: Root locus of the internal loop at v = 50 m·s−1.

The Figure 5.8 shows a diagram of signal flow for this type of regulation.

1

Ref YawRate

2

YawRate

1

SteeringAngle

3

alphaF

4

alphaR

-K-P

1s

Integrator

-K-

Gain3

-K-

Gain4

Saturation

I

P

Figure 5.8: Diagram of augmented yaw rate tracking control with the addition of tire’sslip angles sum control.

5.4 Conclusion

Proposed controllers stabilize a laterally unstable vehicle, but they can also be used

for stable cars. These control techniques are not optimal and robust. There are pos-

sibilities of increasing the quality of a regulation: using robust regulation methods

(proposed regulators suffer on precise of the linearization); creating a batch of con-

trollers for different velocities; there is a possibility of using state feedback and based

on this approach regulation techniques.

Also, for the future work, there are other possibilities of control, which are not

described in this thesis: reference tracking in outer loop for tire’s slip angles αf or

αr, or side-slip angle β. Moreover, there is an idea to work with tire’s slip difference

αf − αr.

24

6. Simulation Tests

6.1 Yaw Rate Tracking Test

Yaw rate tracking test branch was the primary test for all regulators before the test

on a simulator. It was provided by a non-linear single-track model (Eq. 3.6) with pa-

rameters of a vehicle listed under the Figure 4.4. This test branch includes a reference

tracking on different velocities and with added noise imitating wind and real road.

0 5 10 15 20 25 30-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Yaw Rate Tracking & Side-Slip DampingYaw Rate TrackingYaw Rate Tracking & (

f +

r) Damping

Reference

Figure 6.1: Regulator comparison at v = 10 m·s−1.

The Figure 6.1 shows a comparison of all regulators on a sufficiently small velocity,

at v = 10 m·s−1. All of the proposed control techniques work well at a low speed.

With a higher (Figure 6.2) velocity (v = 20 m·s−1) the main difference between the 3rd

controller, which uses the sum of tires’ slip angles damper, and others is correctly seen.

This regulator keeps side-slip sufficiently small to pretend to get a spin (Figure 6.3).

The first two regulators, yaw rate tracker and its augmentation with side-slip reference

tracking work well. Nevertheless, as it can be seen in Figure 6.4, its primary problem

is that on a non-linear model this regulator cannot prevent a spin situation, that is

shown in the next test with a bit higher velocity (v = 23.5 m·s−1). After keeping on a

high yaw rate reference for a sufficiently long period of time, vehicles got to spin. In

comparison, the 3rd controller prevened this situation.

The next Figure 6.5 shows test results including a noise added on side-slip imitating

small hummocks on the road and small wind gusts acting on the vehicle. It can be seen

that the 3rd regulator does not have an excellent noise suppression. It can be solved

by using of a low-pass filter on the inner closed loop, which also brings additional small

control delay.

25

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

Yaw Rate Tracking & Side-Slip DampingYaw Rate TrackingYaw Rate Tracking & (

f +

r) Damping

Reference

Figure 6.2: Regulator comparison at v = 20 m·s−1.

0 5 10 15 20 25 30-0.8

-0.6

-0.4

-0.2

0

0.2

Yaw Rate Tracking & Side-Slip DampingYaw Rate TrackingYaw Rate Tracking & (

f +

r) Damping

Figure 6.3: Regulator comparison at v = 20 m·s−1 (Side-slip).

0 5 10 15 20 25 30-2

-1

0

1

2

3

Yaw Rate Tracking & Side-Slip DampingYaw Rate TrackingYaw Rate Tracking & (

f +

r) Damping

Reference

Figure 6.4: Regulator comparison at v = 23.5 m·s−1.

26

0 5 10 15 20 25 30-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Yaw Rate Tracking & Side-Slip DampingYaw Rate TrackingYaw Rate Tracking & (

f +

r) Damping

ReferenceYaw Rate Tracking & (

f +

r) Damping (Filtered)

Figure 6.5: Regulator comparison with noise at v = 15 m·s−1.

Provided tests correctly show that laterally unstable vehicle can be stabilizable, but

with some restrictions. In chapter [Laterally Unstable Vehicle] was stated that, ac-

cording to the linearized model, these types of control techniques could not destabilize

an unstable vehicle. But, as it seen, tests provided on a non-linear model shows that

there always are some restrictions on a possible physical maximum perfomance of an

unstable car. And this restrictions can be solved, for example by using the 3rd regula-

tor, or by introducing of a non-linear analog of the basis regulator.

Nevertheless, results of tests provided on the simulator are optimistic, and they

show a possibility of control an unstable vehicle and open a wilder field to discussions.

6.2 Spoiler Elimination

Spoiler or rear wing is used to help a driver handle the car at higher velocities.

It produces an anti-lift force that increases the vertical force acting on the rear axle

to improve traction and maneuverability (increasing of lateral forces acting on rear

tires (Eq. 3.23)). On the other hand, cars equipped with rear wing have higher at-

mospheric drag, what increase fuel consumption. Creating a rear wing is always a

trade-off between generated anti-lift and drag forces. But spoilers can be reduced when

a vehicle is equipped with active artificial stabilization. This statement is proven below.

To provide an experimental prove the non-linear single-track model must be aug-

mented by aerodynamic forces acting on a vehicle. These forces can be calculated using

the following equation [1]:

F = cSρ

2v2, (6.1)

where F is a calculating force (Fl for anti-lift or Fd for drag); c is an aerodynamic con-

27

stant (cl for anti-lift or cd for drag); S is an effective cross-sectional area (is assumed

to be constant); ρ is the air density; v is an air velocity (is assumed to be equal to the

vehicle’s velocity).

The augmented non-linear single-track model takes the following form:

β = −ψ + 1mv (cosβ(Fyr(Fzr + Fl) + cos δFyf )− sinβ(Fxr − Fd − sin δFyf )),

v = 1m(sinβ(Fyr(Fzr + Fl) + cos δFyf ) + cosβ(Fxr − Fd − sin δFyf )),

ψ = 1Iz (lf cos δFyf − lrFyr(Fzr + Fl)),

(6.2)

where lateral force acting on a rear wheel Fyr(Fzr +Fl) is now a function of the sum of

vertical forces acting on the rear axle (gravity and anti-lift). Gravity force is constant;

only anti-lift force is changed. Also, drag is a dissipative force acting on a rear axle in

a longitudinal direction.

For this test, the same slightly unstable car is used (with parameters listed under

the Figure 4.4). Three cars are tested: the vehicle without stabilization, the vehicle

equipped with yaw rate tracking controller, and the vehicle equipped with the rear

wing. Coefficients cd (0.36 for car with rear wing and 0.31 for cars without it) and cl

(0.26 only for car with rear wing) are taken from [17]. S is assumed to be 1 (in case of

rear wing) and 8 (in case of a whole car).

The Figures 6.6 and 6.7 show the comparison between yaw rate and additional force

applied by cruise control on vehicle models in a test with applied step with attitude

0.02 rad on side-slip angle β, what imitates a bump or wind gust. It can be correctly

seen that the vehicle equipped with controller keeps zero yaw rate better than the car

equipped with the rear wing. Moreover, the reduced drag also reduces the force applied

to the rear axle to keep the desired velocity.

0 1 2 3 4 5 6 7-0.05

0

0.05

Non-stabilizableYaw Rate Tracking ControlUsed Rear Wing

Figure 6.6: Comparison of a yaw rate of a vehicle with applied wind gust at a velocityv = 40 m·s−1.

28

0 1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

Non-stabilizableYaw Rate Tracking ControlUsed Rear Wing

Figure 6.7: Comparison of applied longitudinal force on the rear wheel by cruise controlwith applied wind gust at velocity v = 40 m·s−1.

29

7. Slip Estimation Algoritms

This chapter is a short overview of a possible measure and estimation techniques

that can be used to get a missing parameters’ values for proposed control architectures:

tire’s slip angles and side-slip angle.

7.1 Overview of Tire’s Slip Angle Estimators

This section describes primary principles of tire’s slip angle estimation. The overview

is based on dissertation work by Lukas Haffner [7].

The measurement of tire’s slip angle α can be done indirectly by velocity sensors in

lateral and longitudinal directions and then applying formulas from vehicle’s kinematics

(Eq. 3.7 and 3.8 with substituted ratio of longitudinal vx and lateral vy velocities for

vehicle’s side-slip angle β):

αf = δ − vyvx−lf ψ

vx, (7.1)

αr = −vyvx

+lrψ

vx. (7.2)

This solution is expensive due to a price of velocity sensors, and it is not suitable

for mass car production. As an alternative, there exist observer methods.

Tire model

Lateral velocityestimation  Slip angle calculation 

δ

α

Fy

vyvx

ψ˙

Figure 7.1: Diagram of tire’s slip angle estimation

Commonly, they are based on separation principle shown in the Figure 7.1. The

estimation is divided into three parts:

• Calculation of lateral tire force Fy by a suitable tire model. Depending on the

30

desired accuracy, models described in the chapter [Used Vehicle Modelling

Approaches] or in [7] can be chosen.

• The second block in the diagram refers to a suitable estimation algorithm for

reaching a value of a lateral velocity vx. For estimation there exist two main

groups of techniques: Kalman filter (Linear Kalman filter, Extended Kalman fil-

ter), and sliding mode observers (such as Utkin observer or Walcott-Zak observer).

A comparison of the mentioned estimators is provided in [7].

• The last block in the diagram refers to α-calculation using kinematics formulas

(Eq. 7.1 and 7.2).

7.2 Overview of Side-Slip Angle Estimators

In vehicle’s side-slip angle measurements there are the same possibilities as well as

issues as in the field of tire’s slip angle estimation. Namely, there is the same possibility

of indirect calculation of the side-slip β from measurements of velocity sensors in lateral

and longitudinal directions with applying of kinematic equation [6] and [7]:

β = arctanvxvy≈ vxvy. (7.3)

But the same problem arises: velocity sensors are expensive. From this fact follows

the same solution: use a side-slip estimator.

There exist hundreds of research articles about the estimation of vehicle’s side-slip

angle, from vehicle dynamics model to neural networks. This overview is based on

the idea described in [15]. Generally, the side-slip estimator is implemented via tires’

slip angles estimation algorithms. The diagram in Figure [7.2] shows commonly used

separation signal flow:

• The first block, tires’ slip angles estimator means the estimator described above

in the section [Overview of Tire’s Slip Angle Estimators]. The output of

this block is tires’ slip angles and lateral velocity of each wheel.

• The second block represents tires’ slip ratios calculation using the formula (Eq.

7.4).

• The third block calculates lateral and longitudinal forces generated on each wheel

via suitable tire’s model. Slip ratio λ of each tire is used to calculate longitudinal

forces generated by that wheel, and slip angle α is used for calculation of lateral

force.

• The last block uses the internal state and its inputs to estimate by suitable method

(as an example Extended Kalman filter) the vehicle’s side-slip angle β. The used

formula can be taken from the nonlinear single-track model. (first equation from

Eq. 3.6)

31

Tire's slip angleestimator 

Tire's slip ratiocalculation 

Tire model 

Side­slipobserver 

vx

δ

ψ˙

vy

ωf

ωr

αf

αr

λf

λr

Fxf

Fxr

Fyf

Fyr

β

Figure 7.2: Diagram of vehicles’s side-slip angle estimation

Tire’s slip ratio can be calculated by the following formula taken from [1, chapter

7]:

λ =|v − ωr|

max(v, ωr), λ ∈ [0, 1], (7.4)

where v ≈ vx + vy is a velocity of the wheel center point, ω is an angular velocity of

the wheel (typically is measured by rotation sensors), r is a wheel radius.

32

8. Force Feedback

The primary task of the whole thesis and the proposed control techniques above is

to stabilize laterally unstable cars or improve their lateral characteristics. But that all

is done by the cost of a real feeling of the tires and road of the traditional steering wheel

because the steering wheel in these control approaches only introduces a reference but

not the actual steering angle set on the front axle via the shaft. In vehicles, steered by

regulators, are often used electronic (or electro-hydraulic) controllers instead of tradi-

tional hydraulic shafts, and these type systems are called steer-by-wire [9] car.

8.1 Overview of Steer-by-Wire Vehicles

The idea of using the steer-by-wire ground vehicles has the following profits for

automotive companies and control engineers [10]:

• an open possibility of using new intelligent software and sensors feedback (as

regulators, proposed in this work);

• modularity of the steering system so it can be adapted for different ground vehi-

cles;

• lower manufacturing costs due to the reduction of used components (in case of a

fully electronic steering column).

These type cars have the same problem as it was in avionics after the introduction

of the fly-by-wire systems. How to create a drive-by-wire vehicle ”feelable” like a tra-

ditional one? What forces and what feedback should the driver feel? Should he or she

feel the actions caused by the regulator? Or should some next regulator be applied on

the steering wheel to control the torque applied on the wheel by the force feedback? Or

should it be done just by a combination of the measured and estimated values? While

searching an answer to these questions we undoubtedly should focus on the avionics’

force feedback solution.

8.2 The Implementation Used in This Thesis

There are many different approaches to generate the force feedback to the steer-

ing wheel starting from an option with augmented existing hydraulic shafts by pump

controlled with the use of a load pressure feedback [10], finishing with an electric drive

engine inside a steering wheel, which generates torque as a result of a combination of

a feedback from the steering system model, a feedback from the vehicle dynamics and

a filtered feedforward input signals taken from the stering wheel itself [9].

33

Simulator in this work implementes force feedback from the aligning torque (Figure

3.5) acting on the front tires in case of δ 6= 0. This is a part of the implementation

described by Paul Yih and J. Christian Gerdes in [9].

The paper referenced above has a good description of the aligning moment:”While a

vehicle is turning, tire forces acting on the steering system tend to resist steering motion

away from the straight-ahead position. These forces can be treated as a disturbance

on the steering system and are directly attributable to tire self-aligning moment”. But

in this paper authors use the torque Mz as a function of the lateral force acting on

tires Fy. This thesis’ upgrade was to use the Pacejka Magic formula to calculate this

torque directly from the tire slip angle. The used formula is discussed in chapter [Used

Vehicle Modelling Approaches].

The used feedback is implemented as it is shown in Figure 8.1. The system’s inputs

are steering angle δ and force acting on the rear axle along the x axis in the vehicle

coordinate system (Figure 3.1). They are used in the vehicle’s model to calculate the

front tire slip angle αf and then the Pacejka Magic formula is applied to estimate the

torque around the z axis in tire coordinate system (Figure 8.2). After that, the torque

goes through a low-pass filter and then it is scaled, saturated and then applied on the

electromotor inside the steering wheel.

The Figure 8.3 presents an example of the torque output together with yaw rate

and the front tire’s slip angle.

Fxr

Steering angle projection + Vehicle's dynamics and

Kinematics

δ

Tire modelαf Mz

Scalling &Saturation

τs

Low­pass filter 

Figure 8.1: Implemented force feedback

34

Z

XY

Mz

Figure 8.2: The tire coordinate system

-0.05

0

0.05

-1

0

1

5 10 15 20-200

0

200s

Mz

Figure 8.3: Example of the applied torque

35

9. Results

All of the goals, listed in chapter [Objectives], are done. Namely:

• Non-linear single-track model is created. Description of the used model is pro-

vided in chapter [Used Vehicle Modelling Approaches].

• The laterally unstable vehicle is defined in chapter [Laterally Unstable Vehi-

cle].

• Implemented single-track model is connected with free simulator Panthera made

by Cruden [20].

• Possible control architectures for laterally unstable ground vehicles are described

in chapter [Proposed Control Architectures].

• Simulation tests are provided in chapter [Simulation Tests].

• Estimation algorithms for missing parameters are overviewed in chapter [Slip

Estimation Algoritms].

• Additionally, a short overview of steer-by-wire vehicles is done in the chapter

[Force Feedback].

36

10. Conclusion

This work completely answers a question named on the begging of this thesis:

”What if we create a laterally unstable ground vehicle?” It has as its pros so its cons,

namely, the main advantage is that unstable ground vehicle is agiler than stable (as it

is in avionics), it is stabilizable, that is shown by provided simulation tests. Also, ac-

tive stabilization can replace a rear wing used for passive vehicle stabilization at higher

speeds.

But there are, as always, some cons. The laterally unstable ground vehicle should

be equipped with the active steering system, the typical driver cannot stabilize this

system by her or himself. Also, there is a necessity of using estimation algorithms for

missing parameters for regulation and using force feedback to stay the car accustomed

for the driver. But everything has a price. In avionics, this price was paid. What stops

us to pay this price in automotive?

Despite the results achieved, this work opens a new field to the discussion. In the

future work, there is a necessity to test proposed control techniques on a high-fidelity

model (twin-track); research of other possible control techniques, which are not connect-

ed with yaw rate tracking; investigate a possibility of using state feedback regulation

methods; discuss control techniques which guarantee the robustness of the stabilization.

All of the possibilities and goals listed above I am going to investigate and solve

during my future Ph.D. education.

37

11. Bibliography

[1] Schramm, D., Hiller, M., & Bardini, R. (2014). Vehicle dynamics. Modeling andSimulation. Berlin, Heidelberg, 151.

[2] Ackermann, J. (1994). Robust decoupling, ideal steering dynamics and yaw sta-bilization of 4WS cars. Automatica, 30 (11), 1761-1768.

[3] Guvenc, B. A., Guvenc, L., & Karaman, S. (2009). Robust yaw stability con-troller design and hardware-in-the-loop testing for a road vehicle. IEEE Transactionson Vehicular Technology, 58 (2), 555-571.

[4] Wang, L., & Ackermann, J. (1998, June). Robustly stabilizing PID controllersfor car steering systems. In American Control Conference, 1998. Proceedings of the1998 (Vol. 1, pp. 41-42). IEEE.

[5] Sienel, W. (1997, December). Estimation of the tire cornering stiffness and itsapplication to active car steering. In Decision and Control, 1997., Proceedings of the36th IEEE Conference on (Vol. 5, pp. 4744-4749). IEEE.

[6] Mondek M. (2018) Active torque vectoring systems for electric drive vehicles(Master thesis). Retrived from Czech Technical University Digital Library

[7] Haffner L. (2008) Real-time tire models for lateral vehicle state estimation (Doc-toral dissertation). Retrived from Universitatsbibliothek der TU Wien

[8] Pacejka, H. (2005). Tire and vehicle dynamics. Elsevier.

[9] Yih, P., & Gerdes, J. C. (2005). Modification of vehicle handling characteristicsvia steer-by-wire. IEEE Transactions on Control Systems Technology, 13 (6), 965-976.

[10] Haggag, S., Alstrom, D., Cetinkunt, S., & Egelja, A. (2005). Modeling, con-trol, and validation of an electro-hydraulic steer-by-wire system for articulated vehicleapplications. IEEE/AsME Transactions on Mechatronics, 10 (6), 688-692.

[11] Ackermann, J. (2012). Robust control: Systems with uncertain physical param-eters. Springer Science & Business Media.

[12] Takahashi, T. (2003). Modeling, analysis and control methods for improvingvehicle dynamic behavior (overview). R&D Review of Toyota CRDL, 38 (4), 1-9.

[13] SAE Surface Vehicle Recommended Practice, ”Vehicle Dynamics Terminology,”SAE Standard J670, Rev. Aug. 2008

[14] Vilela, D., & Barbosa, R. S. (2011). Analytical models correlation for vehicledynamic handling properties. Journal of the Brazilian Society of Mechanical Sciencesand Engineering, 33 (4), 437-444.

[15] Baffet, G., Charara, A., & Lechner, D. (2009). Estimation of vehicle sideslip,

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tire force and wheel cornering stiffness. Control Engineering Practice, 17 (11), 1255-1264.

[16] Whitford, R. (1995, September). A fighter design chronology. In Aircraft En-gineering, Technology, and Operations Congress (p. 3920).

[17] Sadettin Hamut, H., Salah El-Emam, R., Aydin, M., & Dincer, I. (2014). Ef-fects of rear spoilers on ground vehicle aerodynamic drag. International Journal ofNumerical Methods for Heat & Fluid Flow, 24 (3), 627-642.

[18] Skogestad, S., & Postlethwaite, I. (2007). Multivariable feedback control: anal-ysis and design (Vol. 2, pp. 359-368). New York: Wiley.

[19] Milliken, W. F., Milliken, D. L., Metz, L. D., & Kasprzak, E. M. (1997). RaceCar Vehicle Dynamics Book and Problems, Answers and Solutions Set. Warrendale,PA: Society of Automotive Engineers, 1997. 990.

[20] Cruden official web-site: http://www.cruden.com/panthera-software/

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List of Figures

3.1 The vehicle coordinate system . . . . . . . . . . . . . . . . . . . . . . . . 63.2 The single-track model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 The block diagram of the single-track model . . . . . . . . . . . . . . . . 93.4 Dependence of generated lateral force Fy on tire slip-angle α with differ-

ent loads Fz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Dependence of generated aligning moment Mz on tire slip-angle α with

different loads Fz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Compirison of Pacejka and Two-Lines tire models . . . . . . . . . . . . . 12

4.1 Comparison of understeering and oversteering car. Images are takenfrom https://en.wikipedia.org. Green path is an optimal cornering tra-jectory. Understeer: the car goes not turn enough and leave the road.Oversteer: the car turns more sharply and could get to spin. . . . . . . . 14

4.2 Understeer and oversteer definition . . . . . . . . . . . . . . . . . . . . . 144.3 Poles location of Eq. 3.20 for understeering car depended on different

vehicle’s velocities. Arrows show the direction of poles allocation withhigher velocity. Car’s parameters: lf = 1.11 m, lr = 1.89 m, Cf =138820 N·rad−1, Cr = 236620 N·rad−1, m = 1190 kg, Iz = 1141 kg·m−2. 15

4.4 Poles location of Eq. 3.20 for oversteering car depended on differentvehicle’s velocities. Arrows show the direction of poles allocation withhigher velocity. Car’s parameters: lf = 2.07 m, lr = 0.93 m, Cf =258700 N·rad−1, Cr = 116730 N·rad−1, m = 1190 kg, Iz = 3900 kg·m−2.In this case vcrit ≈ 23 m·s−1. . . . . . . . . . . . . . . . . . . . . . . . . 16

4.5 Paths of the same vehicle on the different velocities, steering angle δ =0.2 rad. Cars are plotted every 1.5 s. Car’s parameters are the same asin the Figure 4.4. In this case vcrit ≈ 23 m·s−1. (For the path generationa plotting function created by M.Mondek in [6] was used.) . . . . . . . . 16

4.6 Tires’ slip angles during cornering maneuver v = 25 m s−1, δ = 0.08 rad(≈ 4.5 deg) of unstable vehicle with parameters as in Figure 4.4 . . . . . 17

4.7 Tires’ slip angles during cornering maneuver v = 10 m s−1, δ = 0.08 rad(≈ 4.5 deg). Vehicle parameters are the same as in Figure 4.4 . . . . . . 18

4.8 Yaw rate of the vehicle with m = 1190 kg, L = lf + lr = 3 m. Steeringangle δ = 0.01 rad., velocity v = 30 m·s−1. Cars with GC (gravity center)equals to 37, and 42 percent are understeering; 50 - neutral steering; and55 - oversteering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.1 Diagram of yaw rate tracking control. . . . . . . . . . . . . . . . . . . . 205.2 Root locus of the closed system at v = 50 m·s−1. . . . . . . . . . . . . . 215.3 Changing of pole location by side-slip damper for unstable car at v = 50

m·s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4 Diagram of augmented yaw rate tracking control with the addition of

side-slip and steering angles control. . . . . . . . . . . . . . . . . . . . . 225.5 The behavior of the regulated unstable car equipped with yaw rate track-

ing with tire’s slip sum damper on velocity v = 50 m·s−1. For higheryaw rate references artificial boundness arises. . . . . . . . . . . . . . . . 23

5.6 The behavior of the regulated unstable car equipped with yaw rate track-ing with tire’s slip sum damper on velocity v = 20 m·s−1. There is nobounding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.7 Root locus of the internal loop at v = 50 m·s−1. . . . . . . . . . . . . . . 24

40

5.8 Diagram of augmented yaw rate tracking control with the addition oftire’s slip angles sum control. . . . . . . . . . . . . . . . . . . . . . . . . 24

6.1 Regulator comparison at v = 10 m·s−1. . . . . . . . . . . . . . . . . . . . 256.2 Regulator comparison at v = 20 m·s−1. . . . . . . . . . . . . . . . . . . . 266.3 Regulator comparison at v = 20 m·s−1 (Side-slip). . . . . . . . . . . . . 266.4 Regulator comparison at v = 23.5 m·s−1. . . . . . . . . . . . . . . . . . . 266.5 Regulator comparison with noise at v = 15 m·s−1. . . . . . . . . . . . . 276.6 Comparison of a yaw rate of a vehicle with applied wind gust at a velocity

v = 40 m·s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.7 Comparison of applied longitudinal force on the rear wheel by cruise

control with applied wind gust at velocity v = 40 m·s−1. . . . . . . . . . 29

7.1 Diagram of tire’s slip angle estimation . . . . . . . . . . . . . . . . . . . 307.2 Diagram of vehicles’s side-slip angle estimation . . . . . . . . . . . . . . 32

8.1 Implemented force feedback . . . . . . . . . . . . . . . . . . . . . . . . . 348.2 The tire coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . 358.3 Example of the applied torque . . . . . . . . . . . . . . . . . . . . . . . 35

41